Journal of the Operations Research Society of China ›› 2025, Vol. 13 ›› Issue (1): 83-113.doi: 10.1007/s40305-023-00458-4

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  • 收稿日期:2022-03-22 出版日期:2025-03-30 发布日期:2025-03-20
  • 通讯作者: Hui Zhang,Yu-Hong Dai E-mail:h.zhang1984@163.com;dyh@lsec.cc.ac.cn
  • 基金资助:
    The first author was supported by the National Natural Science Foundation of China (No. 11971480), the Natural Science Fund of Hunan for Excellent Youth (No. 2020JJ3038), and the Fund for NUDT Young Innovator Awards (No. 20190105). The second author was supported by the National Natural Science Foundation of China (Nos. 11991020, 11631013, 12021001, 11971372 and 11991021) and the Strategic Priority Research Program of Chinese Academy of Sciences (No. XDA27000000).

Mirror Frameworks for Relatively Lipschitz and Monotone-Like Variational Inequalities

Hui Zhang1, Yu-Hong Dai2   

  1. 1 Department of Mathematics, National University of Defense Technology, Changsha 410073, Hunan, China;
    2 School of Mathematical Sciences, Chinese Academy of Sciences, Beijing 100049, China
  • Received:2022-03-22 Online:2025-03-30 Published:2025-03-20
  • Contact: Hui Zhang,Yu-Hong Dai E-mail:h.zhang1984@163.com;dyh@lsec.cc.ac.cn

Abstract: Nonconvex-nonconcave saddle-point optimization in machine learning has triggered lots of research for studying non-monotone variational inequalities (VIs). In this work, we introduce two mirror frameworks, called mirror extragradient method and mirror extrapolation method, for approximating solutions to relatively Lipschitz and monotone-like VIs. The former covers the well-known Nemirovski’s mirror prox method and Nesterov’s dual extrapolation method, and the recently proposed Bregman extragradient method; all of them can be reformulated into a scheme that is very similar to the original form of extragradient method. The latter includes the operator extrapolation method and the Bregman extrapolation method as its special cases. The proposed mirror frameworks allow us to present a unified and improved convergence analysis for all these existing methods under relative Lipschitzness and monotone-like conditions that may be the currently weakest assumptions guaranteeing (sub) linear convergence.

Key words: Extragradient, Extrapolation, Bregman distance, Saddle-point, Mirror descent, Relative Lipschitzness, Monotone, Variational inequality

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